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Theorem eqglact 17853
Description: A left coset can be expressed as the image of a left action. (Contributed by Mario Carneiro, 20-Sep-2015.)
Hypotheses
Ref Expression
eqger.x 𝑋 = (Base‘𝐺)
eqger.r = (𝐺 ~QG 𝑌)
eqglact.3 + = (+g𝐺)
Assertion
Ref Expression
eqglact ((𝐺 ∈ Grp ∧ 𝑌𝑋𝐴𝑋) → [𝐴] = ((𝑥𝑋 ↦ (𝐴 + 𝑥)) “ 𝑌))
Distinct variable groups:   𝑥, +   𝑥,   𝑥,𝐺   𝑥,𝑋   𝑥,𝐴   𝑥,𝑌

Proof of Theorem eqglact
Dummy variable 𝑔 is distinct from all other variables.
StepHypRef Expression
1 eqger.x . . . . . . 7 𝑋 = (Base‘𝐺)
2 eqid 2771 . . . . . . 7 (invg𝐺) = (invg𝐺)
3 eqglact.3 . . . . . . 7 + = (+g𝐺)
4 eqger.r . . . . . . 7 = (𝐺 ~QG 𝑌)
51, 2, 3, 4eqgval 17851 . . . . . 6 ((𝐺 ∈ Grp ∧ 𝑌𝑋) → (𝐴 𝑥 ↔ (𝐴𝑋𝑥𝑋 ∧ (((invg𝐺)‘𝐴) + 𝑥) ∈ 𝑌)))
6 3anass 1080 . . . . . 6 ((𝐴𝑋𝑥𝑋 ∧ (((invg𝐺)‘𝐴) + 𝑥) ∈ 𝑌) ↔ (𝐴𝑋 ∧ (𝑥𝑋 ∧ (((invg𝐺)‘𝐴) + 𝑥) ∈ 𝑌)))
75, 6syl6bb 276 . . . . 5 ((𝐺 ∈ Grp ∧ 𝑌𝑋) → (𝐴 𝑥 ↔ (𝐴𝑋 ∧ (𝑥𝑋 ∧ (((invg𝐺)‘𝐴) + 𝑥) ∈ 𝑌))))
87baibd 529 . . . 4 (((𝐺 ∈ Grp ∧ 𝑌𝑋) ∧ 𝐴𝑋) → (𝐴 𝑥 ↔ (𝑥𝑋 ∧ (((invg𝐺)‘𝐴) + 𝑥) ∈ 𝑌)))
983impa 1100 . . 3 ((𝐺 ∈ Grp ∧ 𝑌𝑋𝐴𝑋) → (𝐴 𝑥 ↔ (𝑥𝑋 ∧ (((invg𝐺)‘𝐴) + 𝑥) ∈ 𝑌)))
109abbidv 2890 . 2 ((𝐺 ∈ Grp ∧ 𝑌𝑋𝐴𝑋) → {𝑥𝐴 𝑥} = {𝑥 ∣ (𝑥𝑋 ∧ (((invg𝐺)‘𝐴) + 𝑥) ∈ 𝑌)})
11 dfec2 7903 . . 3 (𝐴𝑋 → [𝐴] = {𝑥𝐴 𝑥})
12113ad2ant3 1129 . 2 ((𝐺 ∈ Grp ∧ 𝑌𝑋𝐴𝑋) → [𝐴] = {𝑥𝐴 𝑥})
13 eqid 2771 . . . . . . . . 9 (𝑔𝑋 ↦ (𝑥𝑋 ↦ (𝑔 + 𝑥))) = (𝑔𝑋 ↦ (𝑥𝑋 ↦ (𝑔 + 𝑥)))
1413, 1, 3, 2grplactcnv 17726 . . . . . . . 8 ((𝐺 ∈ Grp ∧ 𝐴𝑋) → (((𝑔𝑋 ↦ (𝑥𝑋 ↦ (𝑔 + 𝑥)))‘𝐴):𝑋1-1-onto𝑋((𝑔𝑋 ↦ (𝑥𝑋 ↦ (𝑔 + 𝑥)))‘𝐴) = ((𝑔𝑋 ↦ (𝑥𝑋 ↦ (𝑔 + 𝑥)))‘((invg𝐺)‘𝐴))))
1514simprd 483 . . . . . . 7 ((𝐺 ∈ Grp ∧ 𝐴𝑋) → ((𝑔𝑋 ↦ (𝑥𝑋 ↦ (𝑔 + 𝑥)))‘𝐴) = ((𝑔𝑋 ↦ (𝑥𝑋 ↦ (𝑔 + 𝑥)))‘((invg𝐺)‘𝐴)))
1613, 1grplactfval 17724 . . . . . . . . 9 (𝐴𝑋 → ((𝑔𝑋 ↦ (𝑥𝑋 ↦ (𝑔 + 𝑥)))‘𝐴) = (𝑥𝑋 ↦ (𝐴 + 𝑥)))
1716adantl 467 . . . . . . . 8 ((𝐺 ∈ Grp ∧ 𝐴𝑋) → ((𝑔𝑋 ↦ (𝑥𝑋 ↦ (𝑔 + 𝑥)))‘𝐴) = (𝑥𝑋 ↦ (𝐴 + 𝑥)))
1817cnveqd 5435 . . . . . . 7 ((𝐺 ∈ Grp ∧ 𝐴𝑋) → ((𝑔𝑋 ↦ (𝑥𝑋 ↦ (𝑔 + 𝑥)))‘𝐴) = (𝑥𝑋 ↦ (𝐴 + 𝑥)))
191, 2grpinvcl 17675 . . . . . . . 8 ((𝐺 ∈ Grp ∧ 𝐴𝑋) → ((invg𝐺)‘𝐴) ∈ 𝑋)
2013, 1grplactfval 17724 . . . . . . . 8 (((invg𝐺)‘𝐴) ∈ 𝑋 → ((𝑔𝑋 ↦ (𝑥𝑋 ↦ (𝑔 + 𝑥)))‘((invg𝐺)‘𝐴)) = (𝑥𝑋 ↦ (((invg𝐺)‘𝐴) + 𝑥)))
2119, 20syl 17 . . . . . . 7 ((𝐺 ∈ Grp ∧ 𝐴𝑋) → ((𝑔𝑋 ↦ (𝑥𝑋 ↦ (𝑔 + 𝑥)))‘((invg𝐺)‘𝐴)) = (𝑥𝑋 ↦ (((invg𝐺)‘𝐴) + 𝑥)))
2215, 18, 213eqtr3d 2813 . . . . . 6 ((𝐺 ∈ Grp ∧ 𝐴𝑋) → (𝑥𝑋 ↦ (𝐴 + 𝑥)) = (𝑥𝑋 ↦ (((invg𝐺)‘𝐴) + 𝑥)))
2322cnveqd 5435 . . . . 5 ((𝐺 ∈ Grp ∧ 𝐴𝑋) → (𝑥𝑋 ↦ (𝐴 + 𝑥)) = (𝑥𝑋 ↦ (((invg𝐺)‘𝐴) + 𝑥)))
24233adant2 1125 . . . 4 ((𝐺 ∈ Grp ∧ 𝑌𝑋𝐴𝑋) → (𝑥𝑋 ↦ (𝐴 + 𝑥)) = (𝑥𝑋 ↦ (((invg𝐺)‘𝐴) + 𝑥)))
2524imaeq1d 5605 . . 3 ((𝐺 ∈ Grp ∧ 𝑌𝑋𝐴𝑋) → ((𝑥𝑋 ↦ (𝐴 + 𝑥)) “ 𝑌) = ((𝑥𝑋 ↦ (((invg𝐺)‘𝐴) + 𝑥)) “ 𝑌))
26 imacnvcnv 5739 . . 3 ((𝑥𝑋 ↦ (𝐴 + 𝑥)) “ 𝑌) = ((𝑥𝑋 ↦ (𝐴 + 𝑥)) “ 𝑌)
27 eqid 2771 . . . . 5 (𝑥𝑋 ↦ (((invg𝐺)‘𝐴) + 𝑥)) = (𝑥𝑋 ↦ (((invg𝐺)‘𝐴) + 𝑥))
2827mptpreima 5771 . . . 4 ((𝑥𝑋 ↦ (((invg𝐺)‘𝐴) + 𝑥)) “ 𝑌) = {𝑥𝑋 ∣ (((invg𝐺)‘𝐴) + 𝑥) ∈ 𝑌}
29 df-rab 3070 . . . 4 {𝑥𝑋 ∣ (((invg𝐺)‘𝐴) + 𝑥) ∈ 𝑌} = {𝑥 ∣ (𝑥𝑋 ∧ (((invg𝐺)‘𝐴) + 𝑥) ∈ 𝑌)}
3028, 29eqtri 2793 . . 3 ((𝑥𝑋 ↦ (((invg𝐺)‘𝐴) + 𝑥)) “ 𝑌) = {𝑥 ∣ (𝑥𝑋 ∧ (((invg𝐺)‘𝐴) + 𝑥) ∈ 𝑌)}
3125, 26, 303eqtr3g 2828 . 2 ((𝐺 ∈ Grp ∧ 𝑌𝑋𝐴𝑋) → ((𝑥𝑋 ↦ (𝐴 + 𝑥)) “ 𝑌) = {𝑥 ∣ (𝑥𝑋 ∧ (((invg𝐺)‘𝐴) + 𝑥) ∈ 𝑌)})
3210, 12, 313eqtr4d 2815 1 ((𝐺 ∈ Grp ∧ 𝑌𝑋𝐴𝑋) → [𝐴] = ((𝑥𝑋 ↦ (𝐴 + 𝑥)) “ 𝑌))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 382  w3a 1071   = wceq 1631  wcel 2145  {cab 2757  {crab 3065  wss 3723   class class class wbr 4787  cmpt 4864  ccnv 5249  cima 5253  1-1-ontowf1o 6029  cfv 6030  (class class class)co 6796  [cec 7898  Basecbs 16064  +gcplusg 16149  Grpcgrp 17630  invgcminusg 17631   ~QG cqg 17798
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4905  ax-sep 4916  ax-nul 4924  ax-pow 4975  ax-pr 5035  ax-un 7100
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rmo 3069  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4227  df-pw 4300  df-sn 4318  df-pr 4320  df-op 4324  df-uni 4576  df-iun 4657  df-br 4788  df-opab 4848  df-mpt 4865  df-id 5158  df-xp 5256  df-rel 5257  df-cnv 5258  df-co 5259  df-dm 5260  df-rn 5261  df-res 5262  df-ima 5263  df-iota 5993  df-fun 6032  df-fn 6033  df-f 6034  df-f1 6035  df-fo 6036  df-f1o 6037  df-fv 6038  df-riota 6757  df-ov 6799  df-oprab 6800  df-mpt2 6801  df-ec 7902  df-0g 16310  df-mgm 17450  df-sgrp 17492  df-mnd 17503  df-grp 17633  df-minusg 17634  df-eqg 17801
This theorem is referenced by:  eqgen  17855  cldsubg  22134  tgpconncompeqg  22135  snclseqg  22139
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